An Introduction to Elliptic Curves

Elliptic Curves and the Abelian Grou

An Introduction to Elliptic Curves

Elliptic Curves are curves of the form
y2=x3+ax+by^2=x^3+ax+b
This is called a Weierstrass equation. The rational points on this elliptic curve miraculously create an abelian group under an operation called point addition.

Point Addition

Given two points
PP
and
QQ
, we can calculate a third point
P⊕QP \oplus Q
on the elliptic curve itself using a simple algorithm:
The images here are from Joseph Silverman's presentation slides from 2006, which are an excellent reference and can be found here.
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First, we draw the line
LL
through the points
PP
and
QQ
, which intersects the curve at a third point
RR
. We then draw a vertical line passing through
RR
which intersects the curve at a second point we call
R′R^\prime
.
R′R^\prime
is the result of
P⊕QP \oplus Q
, the point addition of
PP
and
QQ
.

Adding a Point to Itself

We can also add a point
PP
to itself, but how can we do that with infinitely many lines passing through? We simply take the tangent to the curve and find the point
RR
it intersects at before mirroring the point to
R′=2PR^\prime = 2P
.

The Point at Infinity

Rarely, you may find yourself adding
PP
to
−P-P
, the point directly below
PP
. In this case there is no third point of intersection and we say the result of point addition is
O\mathcal{O}
, the point "at infinity".

Addition Rules

Last modified 1yr ago